58
GAUGE FIELDS AND STRINGS
If cp(x) -► 0, eq. (4.35) has only the zero continuous solution. Let us
X - * 00
suppose now, that we have introduced a set of vortices, placed at the
points {jc„} with strengths {q„}. This means that we integrate over the
multivalued fields (p which change by 2nq^ as we go around the point
x^. In this sector we do have a nontrivial classical solution of (4.35). It is
given by:
< P = Z «<.
a = 1
Z =
iX2
(4.36)
The formula (4.36) can be viewed as a continuum approximation to the
classical solution, minimizing the action:
x ,6
< P x-^s)
(4.37)
with any periodic u((p):
u((p + 2n) = u((p)
/
- ( 2
^ ^ 1
(4.38)
This solution far from the singularity is universal and described by
(4.36). The structure near
(the core of the vortex) depends on the
detailed form of m, but fortunately it appears to be irrelevant.
Substituting (4.36) into (4.34) we obtain:
-, '1 Z
log :
^ l^a^b
I
R
- -h const,
(4.39)
(R being the size of the system; the second term is the self-energy of the
vortex).
Noticing that (l/2n) log(/^/|jc|) is just the inverse Laplacian of a twodimensional Coulomb energy we see that taking account of instantons
for (4.21) leads to the continuum version of formula (4.30). We see that
the obscure transformations leading to (4.30) serve a simple purpose—
they take account of vortices.
However, in the present case, owing to the “confinement” of vortices
they do not have any qualitative effects (at large j?). In the next section
we shall consider the case when such effects are present.
Let us mention what happens for ^ = 3. In this case we have, instead
of point-like singularities, singular lines. The field (p has a 2tc jump as we